Example: Reed-Muller codes RM q ( m , r ) : = { ( f ( x ) : x ∈ F m q ) , f ∈ F q [ X 1 , . . . , X m ] , deg f ≤ r } c x = f ( x ) Assume r ≤ q − 2 , and let: – c = ( f ( x ) : x ∈ F m q ) ∈ RM q ( m , r ) – φ : F q → F m q affine and injective c | L ⇒ affine line L : = φ ( F q ) ⊂ F m q Then, the restriction of c to L ( or to φ ): L c | L : = (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) F m at coordinate i ∈ F m q Local correction of y ∈ F q : q 1. Pick at random a line L ⊂ F m q such that i ∈ L . 2. Correct y | L as a noisy RS q ( r ) codeword, and output ˜ y i . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 6/33
Example: Reed-Muller codes RM q ( m , r ) : = { ( f ( x ) : x ∈ F m q ) , f ∈ F q [ X 1 , . . . , X m ] , deg f ≤ r } c x = f ( x ) Assume r ≤ q − 2 , and let: – c = ( f ( x ) : x ∈ F m q ) ∈ RM q ( m , r ) – φ : F q → F m q affine and injective c | L ⇒ affine line L : = φ ( F q ) ⊂ F m q Then, the restriction of c to L ( or to φ ): L c | L : = (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) F m at coordinate i ∈ F m q Local correction of y ∈ F q : q 1. Pick at random a line L ⊂ F m q such that i ∈ L . 2. Correct y | L as a noisy RS q ( r ) codeword, and output ˜ y i . RM q ( m , r ) is locally correctable with ℓ = n 1/ m and ε = 2 1 − r / q · δ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 6/33
High-rate construction: lifted codes (1) Issue: if r ≤ q − 2, the rate of RM q ( m , r ) is ≃ ( r / q ) m . m ! J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 7/33
High-rate construction: lifted codes (1) Issue: if r ≤ q − 2, the rate of RM q ( m , r ) is ≃ ( r / q ) m . m ! Idea: consider the set of all polynomials f satisfying the “restriction property”: for every affine line L given by φ , (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) Are there more polynomials than in RM codes? J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 7/33
High-rate construction: lifted codes (1) Issue: if r ≤ q − 2, the rate of RM q ( m , r ) is ≃ ( r / q ) m . m ! Idea: consider the set of all polynomials f satisfying the “restriction property”: for every affine line L given by φ , (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) Are there more polynomials than in RM codes? Example ( q = 4, m = 2, r = 2). f ( X , Y ) = X 2 Y 2 ∈ F 4 [ X , Y ] , hence deg ( f ) = 4 > 2 Affine line L given by φ ( T ) = ( aT + b , cT + d ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 7/33
High-rate construction: lifted codes (1) Issue: if r ≤ q − 2, the rate of RM q ( m , r ) is ≃ ( r / q ) m . m ! Idea: consider the set of all polynomials f satisfying the “restriction property”: for every affine line L given by φ , (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) Are there more polynomials than in RM codes? Example ( q = 4, m = 2, r = 2). f ( X , Y ) = X 2 Y 2 ∈ F 4 [ X , Y ] , hence deg ( f ) = 4 > 2 Affine line L given by φ ( T ) = ( aT + b , cT + d ) ( f ◦ φ )( T ) = ( aT + b ) 2 ( cT + d ) 2 = ( a 2 T 2 + b 2 )( c 2 T 2 + d 2 ) = ( ac ) 2 T 4 + ( ad + bc ) 2 T 2 + ( bd ) 2 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 7/33
High-rate construction: lifted codes (1) Issue: if r ≤ q − 2, the rate of RM q ( m , r ) is ≃ ( r / q ) m . m ! Idea: consider the set of all polynomials f satisfying the “restriction property”: for every affine line L given by φ , (( f ◦ φ )( t ) : t ∈ F q ) ∈ RS q ( r ) Are there more polynomials than in RM codes? Example ( q = 4, m = 2, r = 2). f ( X , Y ) = X 2 Y 2 ∈ F 4 [ X , Y ] , hence deg ( f ) = 4 > 2 Affine line L given by φ ( T ) = ( aT + b , cT + d ) ( f ◦ φ )( T ) = ( aT + b ) 2 ( cT + d ) 2 = ( a 2 T 2 + b 2 )( c 2 T 2 + d 2 ) = ( ac ) 2 T 4 + ( ad + bc ) 2 T 2 + ( bd ) 2 = ( ad + bc ) 2 T 2 + ( ac ) 2 T + ( bd ) 2 mod ( T 4 − T ) ⇒ for every φ , the “restriction” ( f ◦ φ )( T ) can be interpolated as a univariate polynomial of degree ≤ 2 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 7/33
High-rate construction: lifted codes (2) ◮ A m : = F m q ) ∈ F A m ev A m ( f ) : = ( f ( x ) : x ∈ F m q q ◮ Emb A ( m ) : = { φ : F q → F m q , injective and affine } Definition (lifted Reed-Solomon code [GKS13] reformulated). Lift ( RS q ( r ) , m ) : = { ev A m ( f ) , f ∈ F q [ X ] | ∀ φ ∈ Emb A ( m ) , ev A 1 ( f ◦ φ ) ∈ RS q ( r ) } J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 8/33
High-rate construction: lifted codes (2) ◮ A m : = F m q ) ∈ F A m ev A m ( f ) : = ( f ( x ) : x ∈ F m q q ◮ Emb A ( m ) : = { φ : F q → F m q , injective and affine } Definition (lifted Reed-Solomon code [GKS13] reformulated). Lift ( RS q ( r ) , m ) : = { ev A m ( f ) , f ∈ F q [ X ] | ∀ φ ∈ Emb A ( m ) , ev A 1 ( f ◦ φ ) ∈ RS q ( r ) } Lift ( RS q ( r ) , m ) is locally correctable with ℓ = n 1/ m and ε = 2 1 − r / q · δ . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 8/33
High-rate construction: lifted codes (2) ◮ A m : = F m q ) ∈ F A m ev A m ( f ) : = ( f ( x ) : x ∈ F m q q ◮ Emb A ( m ) : = { φ : F q → F m q , injective and affine } Definition (lifted Reed-Solomon code [GKS13] reformulated). Lift ( RS q ( r ) , m ) : = { ev A m ( f ) , f ∈ F q [ X ] | ∀ φ ∈ Emb A ( m ) , ev A 1 ( f ◦ φ ) ∈ RS q ( r ) } Lift ( RS q ( r ) , m ) is locally correctable with ℓ = n 1/ m and ε = 2 1 − r / q · δ . What about the dimension/rate? Theorem (characteristic 2, simplified from [GKS13]). For every m ≥ 2 and 0 < R 0 < 1, there exists q > 0 and r ≤ q − 2 such that Lift ( RS q ( r ) , m ) is locally correctable with rate R ≥ R 0 . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 8/33
Rate of lifted codes Bounds in [GKS13] are far from being tight . ◮ Ex: for m = 2 and R 0 = 1/2, GKS theorem requires n = q m ≥ 2 64 . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 9/33
Rate of lifted codes Bounds in [GKS13] are far from being tight . ◮ Ex: for m = 2 and R 0 = 1/2, GKS theorem requires n = q m ≥ 2 64 . Theorem [characteristic 2, finite length n = q 2 = 2 2 e ]. For m = 2, q = 2 e and r = ( 1 − 2 − c ) q − 1, � 3 c − 1 � c � c R = 1 − 5 � 3 + 1 � 1 + 1 � . 4 4 4 4 2 e 2 c + 2 ◮ actually, n = q 2 ≥ 2 6 = 64 is enough to achieve R ≥ 1/2. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 9/33
Degree sets Lifted codes are monomial , i.e. generated by evaluations of monomials 1 . . . X d m ev A m ( X d 1 m ) = ev A m ( X d ) Degree set of a monomial code [GKS13]: Deg ( C ) : = { d ∈ [ 0, q − 1 ] m , ev A m ( X d ) ∈ C} J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 10/33
Degree sets Lifted codes are monomial , i.e. generated by evaluations of monomials 1 . . . X d m ev A m ( X d 1 m ) = ev A m ( X d ) Degree set of a monomial code [GKS13]: Deg ( C ) : = { d ∈ [ 0, q − 1 ] m , ev A m ( X d ) ∈ C} A representation for m = 2: d 2 d 2 d 2 d 1 d 1 d 1 RM 4 ( 2, 4 ) RM 4 ( 2, 2 ) Lift ( RS 4 ( 2 ) , 2 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 10/33
“Fractal” representation of degree sets q = 4, r = 2 q = 8, r = 6 q = 16, r = 14 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 11/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 11/33
Evaluation on projective spaces Projective space P m : = A m + 1 \ { 0 } where a ∼ b iff ∃ λ ∈ F × � � / ∼ q , a = λ b Defining an evaluation map over P m requires: ◮ homogeneous polynomials f ∈ F q [ X ] H v of fixed degree v , ◮ to choose a representative for every u ∈ P m (see [Lac86]): u = ( 0 : · · · : 0 : 1 : ∗ : · · · : ∗ ) ∈ P m We get: f ( u ) : = f ( 0, . . . , 0, 1, ∗ , . . . , ∗ ) ∈ F q ev P m ( f ) : = ( f ( u ) : u ∈ P m ) ∈ F P m q J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 12/33
Projective lifted codes Example. Projective Reed-Solomon code: PRS q ( r ) = { ev P 1 ( f ) = ( f ( x ) : x ∈ P 1 ) , f ∈ F q [ X , Y ] H r } J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 13/33
Projective lifted codes Example. Projective Reed-Solomon code: PRS q ( r ) = { ev P 1 ( f ) = ( f ( x ) : x ∈ P 1 ) , f ∈ F q [ X , Y ] H r } Let Emb P ( m ) : = { φ : F 2 q → F m + 1 linear and injective } . q Definition (lifted projective RS codes). Let v = r + ( m − 1 )( q − 1 ) . Lift ( PRS q ( r ) , m ) : = { ev P m ( f ) , f ∈ F q [ X ] H v | ∀ φ ∈ Emb P ( m ) , ev P 1 ( f ◦ φ ) ∈ PRS q ( r ) } J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 13/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code ◮ are monomial , with an explicit bijection between the degree sets of Lift ( RS q ( r − 1 ) , m ) , Lift ( PRS q ( r ) , m ) and Lift ( PRS q ( r ) , m − 1 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code ◮ are monomial , with an explicit bijection between the degree sets of Lift ( RS q ( r − 1 ) , m ) , Lift ( PRS q ( r ) , m ) and Lift ( PRS q ( r ) , m − 1 ) ◮ satisfy the puncturing/shortening relation 0 → Lift ( RS q ( r − 1 ) , m ) → Lift ( PRS q ( r ) , m ) π → Lift ( PRS q ( r ) , m − 1 ) → 0 , − where π is induced by P m → P m − 1 . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code ◮ are monomial , with an explicit bijection between the degree sets of Lift ( RS q ( r − 1 ) , m ) , Lift ( PRS q ( r ) , m ) and Lift ( PRS q ( r ) , m − 1 ) ◮ satisfy the puncturing/shortening relation 0 → Lift ( RS q ( r − 1 ) , m ) → Lift ( PRS q ( r ) , m ) π → Lift ( PRS q ( r ) , m − 1 ) → 0 , − where π is induced by P m → P m − 1 . ◮ are (up to equivalence) if q − 1 and n = q m + 1 q − 1 are coprime cyclic codes quasi-cyclic codes if q − 1 and gcd ( n , q − 1 ) are coprime n J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code ◮ are monomial , with an explicit bijection between the degree sets of Lift ( RS q ( r − 1 ) , m ) , Lift ( PRS q ( r ) , m ) and Lift ( PRS q ( r ) , m − 1 ) ◮ satisfy the puncturing/shortening relation 0 → Lift ( RS q ( r − 1 ) , m ) → Lift ( PRS q ( r ) , m ) π → Lift ( PRS q ( r ) , m − 1 ) → 0 , − where π is induced by P m → P m − 1 . ◮ are (up to equivalence) if q − 1 and n = q m + 1 q − 1 are coprime cyclic codes quasi-cyclic codes if q − 1 and gcd ( n , q − 1 ) are coprime n ◮ admit many explicit and easily computable information sets J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Main results on projective lifted codes Projective lifted codes... ◮ are locally correctable , with parameters ( ℓ = q + 1, δ , ε = δ / τ ) , where τ is the relative correction capability of the small PRS code ◮ are monomial , with an explicit bijection between the degree sets of Lift ( RS q ( r − 1 ) , m ) , Lift ( PRS q ( r ) , m ) and Lift ( PRS q ( r ) , m − 1 ) ◮ satisfy the puncturing/shortening relation 0 → Lift ( RS q ( r − 1 ) , m ) → Lift ( PRS q ( r ) , m ) π → Lift ( PRS q ( r ) , m − 1 ) → 0 , − where π is induced by P m → P m − 1 . ◮ are (up to equivalence) if q − 1 and n = q m + 1 q − 1 are coprime cyclic codes quasi-cyclic codes if q − 1 and gcd ( n , q − 1 ) are coprime n ◮ admit many explicit and easily computable information sets Details in: Lifted Projective Reed-Solomon Codes , L., DCC, to appear 10.1007/s10623-018-0552-8 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 14/33
Lifted codes when r = q − 2 Remark. Assume r = q − 2. Then, RS q ( q − 2 ) is the parity-check code. q a ∈ RS q ( q − 2 ) ⇐ ∑ a i = 0 ⇒ i = 1 q , ∑ ⇒ ∀ L ⊆ F m c ∈ Lift ( RS q ( q − 2 ) , m ) ⇐ c x = 0 x ∈ L J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 15/33
Lifted codes when r = q − 2 Remark. Assume r = q − 2. Then, RS q ( q − 2 ) is the parity-check code. q a ∈ RS q ( q − 2 ) ⇐ ∑ a i = 0 ⇒ i = 1 q , ∑ ⇒ ∀ L ⊆ F m c ∈ Lift ( RS q ( q − 2 ) , m ) ⇐ c x = 0 x ∈ L A non-full-rank parity-check matrix for Lift ( RS q ( q − 2 ) , m ) : points in F m q ∗ lines in F m 0 0 0 0 · · · · · · · · · indicator vector of line L 1 1 q ∗ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 15/33
Block designs Point-line incidences in the affine space form the affine geometry 2 -design . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 16/33
Block designs Point-line incidences in the affine space form the affine geometry 2 -design . Definition. A t-design of parameters ( n , ℓ , λ ) consists in: ◮ a set X of points, | X | = n , ◮ a set B of blocks B ⊂ X , | B | = ℓ such that every t -set in X appears in exactly λ blocks. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 16/33
Block designs Point-line incidences in the affine space form the affine geometry 2 -design . Definition. A t-design of parameters ( n , ℓ , λ ) consists in: ◮ a set X of points, | X | = n , ◮ a set B of blocks B ⊂ X , | B | = ℓ such that every t -set in X appears in exactly λ blocks. Incidence matrix of a design: points in X ∗ 0 0 0 0 blocks in B · · · · · · · · · indicator vector of block B 1 1 ∗ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 16/33
Codes based on designs, and generalisation The code based on the design D = ( X , B ) is the code C = Code ( D ) ⊆ F X q admitting the incidence matrix of D as a parity-check matrix. Code ( D ) = { c ∈ F X q | ∀ B ∈ B , c | B ∈ Parity } Remark. The dimension of Code ( D ) is highly dependent on the field F q J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 17/33
Codes based on designs, and generalisation The code based on the design D = ( X , B ) is the code C = Code ( D ) ⊆ F X q admitting the incidence matrix of D as a parity-check matrix. Code ( D ) = { c ∈ F X q | ∀ B ∈ B , c | B ∈ Parity } Remark. The dimension of Code ( D ) is highly dependent on the field F q Let F = ( F B ⊆ F B q : B ∈ B ) be a family of codes indexed by blocks B ∈ B . The generalised design-based code based on ( D , F ) is Code ( D , F ) : = { c ∈ F X q | ∀ B ∈ B , c | B ∈ F B } . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 17/33
Design-based codes and LCCs Generalised design-based code C = Code ( D , F ) , where – τ ∈ ( 0, 1 – D be a t - ( n , ℓ + 1, λ ) -design 2 ) is fixed – F = ( F B : B ∈ B ) s.t. every code in F corrects a fraction τ of errors Algorithm. Local correction of y ∈ F X q at i ∈ X ◮ Pick uniformly at random a block B ∈ B such that i ∈ B . ◮ Correct y | B as a noisy codeword from F B , and output ˜ y i . Proposition [ t = 2]. For every δ < τ /2, Code ( D , F ) is a ( ℓ , δ , ε ) -LCC, where ε = δ / τ . √ Proposition [ t = 3]. For every δ < τ − 1/ 2 ℓ , Code ( D , F ) is a ( ℓ , δ , ε ) -LCC where ε = δ ( 1 − δ ) ( τ − δ ) 2 · 1 1 τ 2 ℓ · δ . ℓ ≤ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 18/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 18/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 18/33
Problem statement Given a remote database F ∈ F k q and 1 ≤ i ≤ k , can we retrieve the entry F i , without leaking information on the index i ? J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 19/33
Problem statement Given a remote database F ∈ F k q and 1 ≤ i ≤ k , can we retrieve the entry F i , without leaking information on the index i ? Trivial solution: full download. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 19/33
Problem statement Given a remote database F ∈ F k q and 1 ≤ i ≤ k , can we retrieve the entry F i , without leaking information on the index i ? Trivial solution: full download. Solutions with better communication complexity: ◮ With 1 server, only computational privacy is possible [CGKS95, CG97]. ◮ With ℓ ≥ 2 servers, one can achieve information-theoretic privacy [CGKS95-98]. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 19/33
Definition of PIR [CGKS95] Given a file F and ℓ servers S 1 , . . . , S ℓ , user U wants to recover F i privately. A Private Information Retrieval protocol is a set of algorithms ( Q , A , R ) : J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 20/33
Definition of PIR [CGKS95] Given a file F and ℓ servers S 1 , . . . , S ℓ , user U wants to recover F i privately. A Private Information Retrieval protocol is a set of algorithms ( Q , A , R ) : ( q 1 , . . . , q ℓ ) 1. U generates a query vector q = ( q 1 , . . . , q ℓ ) ← Q ( i ) and sends q j to server S j . . . U S 1 S 2 S ℓ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 20/33
Definition of PIR [CGKS95] Given a file F and ℓ servers S 1 , . . . , S ℓ , user U wants to recover F i privately. A Private Information Retrieval protocol is a set of algorithms ( Q , A , R ) : ( q 1 , . . . , q ℓ ) 1. U generates a query vector q = ( q 1 , . . . , q ℓ ) ← Q ( i ) and sends q j to server S j . . . U 2. Each server S j computes a j = A ( q j , F | S j ) and sends it back to U ( a 1 , . . . , a ℓ ) S 1 S 2 S ℓ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 20/33
Definition of PIR [CGKS95] Given a file F and ℓ servers S 1 , . . . , S ℓ , user U wants to recover F i privately. A Private Information Retrieval protocol is a set of algorithms ( Q , A , R ) : ( q 1 , . . . , q ℓ ) 1. U generates a query vector q = ( q 1 , . . . , q ℓ ) ← Q ( i ) and sends q j to server S j . . . U 2. Each server S j computes a j = A ( q j , F | S j ) and sends it back to U ( a 1 , . . . , a ℓ ) S 1 S 2 S ℓ 3. U recovers F i = R ( q , a , i ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 20/33
Definition of PIR [CGKS95] Given a file F and ℓ servers S 1 , . . . , S ℓ , user U wants to recover F i privately. A Private Information Retrieval protocol is a set of algorithms ( Q , A , R ) : ( q 1 , . . . , q ℓ ) 1. U generates a query vector q = ( q 1 , . . . , q ℓ ) ← Q ( i ) and sends q j to server S j . . . U 2. Each server S j computes a j = A ( q j , F | S j ) and sends it back to U ( a 1 , . . . , a ℓ ) S 1 S 2 S ℓ 3. U recovers F i = R ( q , a , i ) Information-theoretic privacy: I ( i ; q j ) = 0, ∀ j = 1, . . . , ℓ . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 20/33
Motivation Usual goals for PIR: ◮ Low communication complexity ◮ Low storage overhead for the servers ◮ Low computation complexity for algorithms A (server) and R (user) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 21/33
Motivation Usual goals for PIR: ◮ Low communication complexity ◮ Low storage overhead for the servers ◮ Low computation complexity for algorithms A (server) and R (user) Most constructions focus on the download communication complexity – up to the PIR capacity [SJ17] – but require Ω ( k ) computation complexity for each server J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 21/33
Motivation Usual goals for PIR: ◮ Low communication complexity ◮ Low storage overhead for the servers ◮ Low computation complexity for algorithms A (server) and R (user) Most constructions focus on the download communication complexity – up to the PIR capacity [SJ17] – but require Ω ( k ) computation complexity for each server We here focus on the computation complexity , crucial for practicality [OG10]. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 21/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 21/33
Transversal designs A transversal design TD ( ℓ , s ) = ( X , B , G ) is given by: ◮ X a set of points , | X | = n = s ℓ , • • • • • • • • • • • • . . . • • • • • • • • • • • • • • • • J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 22/33
Transversal designs A transversal design TD ( ℓ , s ) = ( X , B , G ) is given by: ◮ X a set of points , | X | = n = s ℓ , G ℓ − 1 G ℓ G 1 G 2 • • • • ◮ groups G = { G j } 1 ≤ j ≤ ℓ satisfying • • • • ℓ ∐ G j and | G j | = s , X = • • • • . . . j = 1 • • • • • • • • • • • • • • • • J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 22/33
Transversal designs A transversal design TD ( ℓ , s ) = ( X , B , G ) is given by: ◮ X a set of points , | X | = n = s ℓ , G ℓ − 1 G ℓ G 1 G 2 • • • • ◮ groups G = { G j } 1 ≤ j ≤ ℓ satisfying • • • • ℓ ∐ G j and | G j | = s , X = • • • • • i j = 1 • • • • ◮ blocks B ∈ B satisfying • • • • • – B ⊂ X and | B | = ℓ ; • • • • • j – for all { i , j } ⊂ X , { i , j } lie: • • • • • either in a single group G ∈ G , or in a unique block B ∈ B J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 22/33
Transversal designs A transversal design TD ( ℓ , s ) = ( X , B , G ) is given by: ◮ X a set of points , | X | = n = s ℓ , G ℓ − 1 G ℓ G 1 G 2 • • • • ◮ groups G = { G j } 1 ≤ j ≤ ℓ satisfying • • • • ℓ ∐ G j and | G j | = s , X = • • • • • i j = 1 • • • • ◮ blocks B ∈ B satisfying • • • • • – B ⊂ X and | B | = ℓ ; • • • • • j – for all { i , j } ⊂ X , { i , j } lie: • • • • • either in a single group G ∈ G , or in a unique block B ∈ B Its incidence matrix (between points and blocks) defines a code. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 22/33
Example The transversal design TD ( 3, 3 ) represented by: G 1 G 2 G 3 B 1 ∪ B 2 ∪ B 3 B = • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • gives an incidence matrix 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 1 H = 0 0 1 0 1 0 1 0 0 1 0 0 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 0 0 1 1 0 0 0 1 0 Its rank over F 3 is 6 the associated code C is a [ 9, 3 ] 3 code. = ⇒ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 23/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 23/33
The PIR scheme Let C ⊆ F n q be a code based on a TD ( ℓ , s ) . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 24/33
The PIR scheme Let C ⊆ F n q be a code based on a TD ( ℓ , s ) . • Initialisation. User U encodes F �→ c ∈ C , and gives c | G j to server S j . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 24/33
The PIR scheme Let C ⊆ F n q be a code based on a TD ( ℓ , s ) . • Initialisation. User U encodes F �→ c ∈ C , and gives c | G j to server S j . • To recover F i = c i , with i ∈ X : 1. User U randomly picks a block B ∈ B containing i . Then U defines: � unique ∈ B ∩ G j if i / ∈ G j q j = Q ( i ) j : = a random point in G j otherwise. 2. Each server S j sends back c q j 3. U recovers c i = − ∑ c q j = − ∑ c b j : i / ∈ G j b ∈ B \{ i } J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 24/33
Privacy and parameters Theorem. This PIR protocol is information-theoretically private. Proof: – the only server which holds F i received a random query; – for each other server S j , query q j gives no information on the block B which has been picked ⇒ no information leaks on i . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 25/33
Privacy and parameters Theorem. This PIR protocol is information-theoretically private. Proof: – the only server which holds F i received a random query; – for each other server S j , query q j gives no information on the block B which has been picked ⇒ no information leaks on i . Features. ◮ communication complexity: ℓ log s uploaded bits, ℓ log q dowloaded bits ◮ computational complexity: ◮ only 1 read for each server (somewhat optimal) ◮ ≤ ℓ additions over F q for the user ◮ storage overhead: ( n − k ) log q bits J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 25/33
Privacy and parameters Theorem. This PIR protocol is information-theoretically private. Proof: – the only server which holds F i received a random query; – for each other server S j , query q j gives no information on the block B which has been picked ⇒ no information leaks on i . Features. ◮ communication complexity: ℓ log s uploaded bits, ℓ log q dowloaded bits ◮ computational complexity: ◮ only 1 read for each server (somewhat optimal) ◮ ≤ ℓ additions over F q for the user ◮ storage overhead: ( n − k ) log q bits Question: transversal designs with good k depending on ( ℓ , s ) ? J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 25/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 25/33
Instances with geometric designs T A , the classical affine transversal design : ◮ X = F m q , m ≥ 2, ◮ G a set of q disjoint hyperplanes partitionning X , ◮ B = { affine lines L secant to each group of G} . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 26/33
Instances with geometric designs T A , the classical affine transversal design : ◮ X = F m q , m ≥ 2, ◮ G a set of q disjoint hyperplanes partitionning X , ◮ B = { affine lines L secant to each group of G} . Proposition. The code based on T A is identical to the code based on the affine geometry design ( i.e. the lifted code with r = q − 2). J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 26/33
Instances with geometric designs T A , the classical affine transversal design : ◮ X = F m q , m ≥ 2, ◮ G a set of q disjoint hyperplanes partitionning X , ◮ B = { affine lines L secant to each group of G} . Proposition. The code based on T A is identical to the code based on the affine geometry design ( i.e. the lifted code with r = q − 2). Instances: – 3.2% storage overhead if #entries ≤ ( #servers ) 2 – 27% storage overhead if #entries ≤ ( #servers ) 3 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 26/33
Instances with geometric designs T A , the classical affine transversal design : ◮ X = F m q , m ≥ 2, ◮ G a set of q disjoint hyperplanes partitionning X , ◮ B = { affine lines L secant to each group of G} . Proposition. The code based on T A is identical to the code based on the affine geometry design ( i.e. the lifted code with r = q − 2). Instances: – 3.2% storage overhead if #entries ≤ ( #servers ) 2 – 27% storage overhead if #entries ≤ ( #servers ) 3 Question: better instances? J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 26/33
Instances with orthogonal arrays An orthogonal array OA ( t , ℓ , s ) of strength t is a list A of words – of length ℓ , – over a finite set S , | S | = s , – such that, for every I ⊂ [ 1, ℓ ] of size t , A | I = S t . Equivalently, an OA ( t , ℓ , s ) is a code A ⊂ S ℓ with dual distance t + 1. S = { a , b } a b b b b a OA ( 2, 3, 2 ) = b a b a a a J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 27/33
Instances with orthogonal arrays An orthogonal array OA ( t , ℓ , s ) of strength t is a list A of words – of length ℓ , – over a finite set S , | S | = s , – such that, for every I ⊂ [ 1, ℓ ] of size t , A | I = S t . Equivalently, an OA ( t , ℓ , s ) is a code A ⊂ S ℓ with dual distance t + 1. S = { a , b } a b b Construction OA → TD : b b a OA ( 2, 3, 2 ) = ◮ X = S × [ 1, ℓ ] b a b a a a ◮ G = { S × { i } , 1 ≤ i ≤ ℓ } ( a , 1 ) ( a , 2 ) ( a , 3 ) ( b , 1 ) ( b , 2 ) ( b , 3 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 27/33
Instances with orthogonal arrays An orthogonal array OA ( t , ℓ , s ) of strength t is a list A of words – of length ℓ , – over a finite set S , | S | = s , – such that, for every I ⊂ [ 1, ℓ ] of size t , A | I = S t . Equivalently, an OA ( t , ℓ , s ) is a code A ⊂ S ℓ with dual distance t + 1. S = { a , b } a b b Construction OA → TD : b b a OA ( 2, 3, 2 ) = ◮ X = S × [ 1, ℓ ] b a b a a a ◮ G = { S × { i } , 1 ≤ i ≤ ℓ } ◮ B = {{ ( c i , i ) , 1 ≤ i ≤ ℓ } , c ∈ OA } ( a , 1 ) ( a , 2 ) ( a , 3 ) ( b , 1 ) ( b , 2 ) ( b , 3 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 27/33
Instances with orthogonal arrays An orthogonal array OA ( t , ℓ , s ) of strength t is a list A of words – of length ℓ , – over a finite set S , | S | = s , – such that, for every I ⊂ [ 1, ℓ ] of size t , A | I = S t . Equivalently, an OA ( t , ℓ , s ) is a code A ⊂ S ℓ with dual distance t + 1. S = { a , b } a b b Construction OA → TD : b b a OA ( 2, 3, 2 ) = ◮ X = S × [ 1, ℓ ] b a b a a a ◮ G = { S × { i } , 1 ≤ i ≤ ℓ } ◮ B = {{ ( c i , i ) , 1 ≤ i ≤ ℓ } , c ∈ OA } ( a , 1 ) ( a , 2 ) ( a , 3 ) ( b , 1 ) ( b , 2 ) ( b , 3 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 27/33
Instances with orthogonal arrays An orthogonal array OA ( t , ℓ , s ) of strength t is a list A of words – of length ℓ , – over a finite set S , | S | = s , – such that, for every I ⊂ [ 1, ℓ ] of size t , A | I = S t . Equivalently, an OA ( t , ℓ , s ) is a code A ⊂ S ℓ with dual distance t + 1. S = { a , b } a b b Construction OA → TD : b b a OA ( 2, 3, 2 ) = ◮ X = S × [ 1, ℓ ] b a b a a a ◮ G = { S × { i } , 1 ≤ i ≤ ℓ } ◮ B = {{ ( c i , i ) , 1 ≤ i ≤ ℓ } , c ∈ OA } ( a , 1 ) ( a , 2 ) ( a , 3 ) ( b , 1 ) ( b , 2 ) ( b , 3 ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 27/33
Resisting collusions Proposition. For t = 2, an OA ( t , ℓ , s ) gives a TD ( ℓ , s ) . J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Resisting collusions Proposition. For t = 2, an OA ( t , ℓ , s ) gives a TD ( ℓ , s ) . Experimentally, for t = 2 and small ℓ and s , codes based on classical affine TDs have the largest dimension. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Resisting collusions Proposition. For t = 2, an OA ( t , ℓ , s ) gives a TD ( ℓ , s ) . Experimentally, for t = 2 and small ℓ and s , codes based on classical affine TDs have the largest dimension. For t ≥ 3, we get TDs such that: for every t -set T of points lying in t different groups, there exists a unique block B ∈ B such that T ⊂ B . ⇒ The PIR protocol resists t − 1 colluding servers. J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Resisting collusions Proposition. For t = 2, an OA ( t , ℓ , s ) gives a TD ( ℓ , s ) . Experimentally, for t = 2 and small ℓ and s , codes based on classical affine TDs have the largest dimension. For t ≥ 3, we get TDs such that: for every t -set T of points lying in t different groups, there exists a unique block B ∈ B such that T ⊂ B . ⇒ The PIR protocol resists t − 1 colluding servers. ◮ OAs with t > 2 exist ( e.g. from Reed-Solomon codes) ◮ But associated TDs lead to codes with poor rates except for t ≪ ℓ J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Resisting collusions Proposition. For t = 2, an OA ( t , ℓ , s ) gives a TD ( ℓ , s ) . Experimentally, for t = 2 and small ℓ and s , codes based on classical affine TDs have the largest dimension. For t ≥ 3, we get TDs such that: for every t -set T of points lying in t different groups, there exists a unique block B ∈ B such that T ⊂ B . ⇒ The PIR protocol resists t − 1 colluding servers. ◮ OAs with t > 2 exist ( e.g. from Reed-Solomon codes) ◮ But associated TDs lead to codes with poor rates except for t ≪ ℓ Details in: Private Information Retrieval from Transversal Designs , L., IEEE TIT, to appear 10.1109/TIT.2018.2861747 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Outline 1. Codes with locality Locality in coding theory, examples Lifted projective Reed-Solomon codes A combinatorial point of view 2. Private information retrieval from transversal designs Private information retrieval (PIR) Transversal designs and codes A new PIR construction Instances 3. Proofs-of-retrievability 4. Conclusion J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 28/33
Proofs-of-retrievability [Juels, Kaliski ’07] Issue: a client wants to verify if a file stored on a server is still retrievable, with a low communication challenge-response protocol “can I get my file?” ? a few bits J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 29/33
Proofs-of-retrievability [Juels, Kaliski ’07] Issue: a client wants to verify if a file stored on a server is still retrievable, with a low communication challenge-response protocol “can I get my file?” ? a few bits Additional constraints: unbounded-use, low client storage, low computation J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 29/33
PoR with lifted codes C = Lift ( RS q ( r ) , m ) Assumption: one can compute independent pseudo-random permutations σ ( κ ) ∈ S ( F q ) , 1 ≤ i ≤ n , κ ∈ K i Initialisation: ◮ User picks κ ∈ K at random ◮ File F is encoded and permuted as follows: F �→ c ∈ C �→ w = σ ( c ) = ( σ ( κ ) 1 ( c 1 ) , . . . , σ ( κ ) n ( c n )) ∈ F n q ◮ User stores κ , server stores w Verification: ◮ User picks a line L ⊂ F m q at random and sends it to the server ◮ Server reads w | L and sends it back to the user ◮ User accepts iff σ − 1 ( w | L ) ∈ RS q ( r ) J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 30/33
Results Informal result (for the lifted code with m = 2): For every ε ≤ ε 0 ≃ 1, we have: the server answers correctly to a fraction ≥ 1 − ε of the challenges, if 1 then with probability ≥ 1 − O the file is extractable from the server. � � n ( ε 0 − ε ) 2 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 31/33
Results Informal result (for the lifted code with m = 2): For every ε ≤ ε 0 ≃ 1, we have: the server answers correctly to a fraction ≥ 1 − ε of the challenges, if 1 then with probability ≥ 1 − O the file is extractable from the server. � � n ( ε 0 − ε ) 2 Details in: New Proofs of Retrievability using Locally Decodable Codes, L. & Levy-dit-Vehel IEEE International Symposium on Information Theory, 2016 J. Lavauzelle – Codes with locality: constructions and applications to cryptographic protocols – PhD defense 31/33
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